Quantum Bayesian Networks: Compositionality and Typing via Linear Logic

2026-04-28

AI Summary

This work addresses the challenge of unifying causal modeling and probabilistic prediction in classical-quantum hybrid systems by proposing a quantum Bayesian network framework grounded in compositional semantics and a type system. The approach introduces linear logic proof nets for the first time to guarantee well-defined compositionality through typing discipline, seamlessly bridging classical Bayesian networks and quantum tensor networks: it recovers standard factorization semantics in purely classical settings and reduces to tensor networks in purely quantum regimes. The resulting formalism is both compositional and type-safe, providing a unified and rigorous semantic foundation for causal inference and probabilistic prediction in hybrid systems.

Abstract

Quantum Bayesian networks provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks-prominent graphical models for classical probabilistic reasoning and inference. Our paper brings compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems.

Problem

• Quantum Bayesian Networks
• Compositionality
• Typing
• Linear Logic
• Causal Relations

Innovation

• Quantum Bayesian Networks
• Compositionality
• Linear Logic
• Proof-nets
• Typing Discipline